University of Mazandaran, Iran
http://cjms.journals.umz.ac.ir
ISSN: 1735-0611
CJMS.4(2)(2015), 189-195
Exact solutions of (3+1)-dimensional nonlinear evolution equations
N. Kadkhoda1
1 Department of Mathematics, Faculty of Basic Sciences, Bozorgmehr
University of Qaenat, Qaenat, Iran
Abstract. In this paper, the kudryashov method has been used for finding the general exact solutions of nonlinear evolution equa-tions that namely the (3 + 1)-dimensional Jimbo-Miwa equation and the (3 + 1)-dimensional potential YTSF equation, when the simplest equation is the equation of Riccati.
Keywords: kudryashov method, Jimbo-Miwa equation, Potential YTSF equation, Riccati equation.
2000 Mathematics subject classification: 35-XX, 35Exx.
1. Introduction
The research area of nonlinear equations has been very active for the past few decades. There are various kinds of nonlinear equations that appear in various areas of physical and mathematical sciences. Much effort has been made on the construction of exact solutions of nonlinear equations, for their important role in the study of nonlinear physical phe-nomena [2, 9]. Nonlinear wave phephe-nomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optimal fiber, biology, oceanology [12], solid state physics, chemical physics and geometry. In recent years, the powerful and efficient methods to find analytic solutions of nonlinear equation have drawn a lot of interest by a diverse group of scientists, such as Tanh-function method, extended
1Corresponding author: [email protected]
Received: 23 March 2015 Revised: 27 August 2015 Accepted: 03 September 2015
Tanh-function method [3, 18, 19], Sine-cosine method [6, 20, 21], (G )-expansion method[1, 7, 13].
In this paper we obtain the exact solutions of Jimbo-Miwa and Po-tential YTSF equations by using the kudryashov method. The Jimbo-Miwa equation is the second equation in the well known KP hierar-chy of integrable systems, which is used to describe certain interesting (3+1)-dimensional waves in physics but not pass any of the conventional integrability tests.
The kudryashov method was developed by Kudryashov [10, 11] on the basis of a procedure analogous to the first step of the test for the Painleve property [4].
The paper is organized as follows:
In Section 2, we explain the main steps of the kudryashov method. In Section 3, we apply this method to the (3 + 1)-dimensional Jimbo-Miwa equation. In Section 4,we use the method to the (3 + 1)-dimensional potential-YTSF equation. concluding remarks are summarized in Sec-tion 5.
2. Description of the kudryashov method
In this section we recall the basic idea of the kudryashov method [8]. Let we have a partial differential equation and by means of an appro-priate transformation this equation is reduced to a nonlinear ordinary differential equation as follow:
P(u, u0, u00, u000, ...) = 0. (2.1)
Exact solution of this equation can be constructed as finite series
u(x) = n
X
i=0
ai(G(x))i (2.2)
Where G(x) is a solution of some ordinary differential equation re-ferred to as the simplest equation, andA0, A1, A2, ..., AM are parameters to be determined.
In this paper we use the equation of Riccati, as the simplest equation
G0(x) =c G(x) +d G(x)2 (2.3)
This equation is well-known nonlinear ordinary differential equation which process exact solution constructed by elementary function. In this paper we work with the following solutions of the Riccati equation
G(x) = cexp [c(x+x0)] 1−dexp [c(x+x0)]
(2.4)
G(x) =− c exp [c(x+x0)] 1 +dexp [c(x+x0)]
(2.5)
for cased >0 ,c <0 , similar above x0 is a constant of integration. the simplest equation has two properties: (i)The order of simplest equa-tion is lesser than equaequa-tion (2.1), (ii) we know the general soluequa-tion of the simplest equation or we know at least exact analytical particular solution(s) of the simplest Eq(2.3).
Nowu(x) can be determined explicitly by using the following three steps: • Step (1). By considering the homogeneous balance between the highest nonlinear terms and the highest order derivatives ofu(x) in Eq.(2.1), the positive integer nin (2.2) is determined.
• Step (2). By substituting Eq.(2.2) with Eq.(2.3) into Eq.(2.1) and collecting all terms with the same powers ofGtogether, the left hand side of Eq.(2.1) is converted into a polynomial. After setting each coefficient of this polynomial to zero, we obtain a set of algebraic equations in terms of Ai(i= 0,1,2, ..., n), c, d. • Step (3). Solving the system of algebraic equations and then
substituting the results and the general solutions of (2.4) or (2.5) into (2.2) gives solutions of (2.1).
Now, we will demonstrate the kudryashov method on two of nonlinear equations, Jimbo-Miwa equation and Potential YTSF equation.
3. The Jimbo-Miwa equation
The Jimbo-Miwa equation is used to describe certain interesting (3+1)-dimensional waves in physics. This equation is
uxxxy+ 3uyuxx+ 3uxuxy+ 2uyt−3uxz= 0. (3.1)
Whereu:Rx×Ry×Rz×R+t →R.
There are many efforts to solve Eq.(3.1). Tang and Liang applied the multi-linear variable separation scheme to Eq.(3.1) [15]. In [5] the Hi-rotas bilinear formalism was employed to obtain three-soliton solutions. some exact solutions of Eq.(3.1) obtained by an extended rational ex-pansion method and symbolic computation[17]. The traveling wave so-lutions for the equation presented using the GG0-expansion method[14]. In this paper we obtain exact solutions of this equation using kudryashov method. By the transformationξ =x+y+z−νt, Eq.(3.1) becomes:
u(4)+ 3[(u0)2]0−(2ν+ 3)u00 = 0 (3.2)
u000+ 3(u0)2−(2ν+ 3)u0= 0 (3.3)
Settingu0=v, Eq.(3.3) becomes
v00+ 3v2−(2ν+ 3)v= 0 (3.4)
With balancing according procedure that be described, we get n = 2, therefore the solution of Eq.(3.4) can be expressed as follows:
v(ξ) = 2
X
i=0
Ai(G(ξ))i (3.5)
whereG(ξ) satisfies the Riccati equation andA0, A1, A2 are parameters to be determined. solutions of Riccati equation are given in (2.4), (2.5). With substituting (3.5) into (3.4) and use of (2.3) and then equating all coefficients of the functionsGi to zero, we obtain A0,A1 and A2:
Case 1 : A0=− c2
3, A1 =−2cd, A2 =−2d
2; 3cd+c3d6= 0(3.6)
Case 2 : A0= 0, A1 =−2cd, A2 =−2d2; −3cd+c3d6= 0(3.7)
Recall u0(x, y, z, t) = v(x, y, z, t), therefore when d < 0 and c > 0 the solution of Eq.(3.1) with using Case 1 (3.6) is given by
u1(x, y, z, t) =−
c2(x+y+z−νt)
3 −
2c
1−dexp[c(x+y+z−νt+x1)] ,
(3.8) whereν =−1
2(c
2+ 3) and x
1 is a constant of integration. also solution of Eq.(3.1) with using Case 2 (3.7) is given by
u2(ξ) =−
2c
1−dexp[c(x+y+z−νt+x2)]
(3.9)
whereν = 12(c2−3) andx
2 is a constant of integration.
And whend >0 andc <0 the solution of Eq.(3.1) with using Case 1 (3.6) is given by
u3(x, y, z, t) =−
c2(x+y+z−νt)
3 −
2c
1 +dexp[c(x+y+z−νt+x3)] ,
(3.10) whereν =−12(c2+ 3) and x3 is a constant of integration. also solution of Eq.(3.1) with using Case 2 (3.7) is given by
u4(x, y, z, t) =−
2c
1 +dexp[c(x+y+z−νt+x4)]
(3.11)
4. The potential-YTSF equation
Now we would like to use this method to obtain the exact solutions of the potential-YTSF equation read:
−4uxt+uxxxz+ 4uxuxz+ 2uxxuz+ 3uyy = 0, (4.1)
Eq.(4.1) is called the potential-YTSF equation which was firstly intro-duced by Yu, Toda, Sasa and Fukuyama (YTSF) [23]. Recently, this equation was studied and some single soliton and periodic solitary so-lutions were obtained [24, 22]. [16] obtained exact soso-lutions of (4.1) by Exp-function method. By the transformationξ =x+y+z−νt, Eq.(4.1) becomes:
u(4)+ 3[(u0)2]0+ (4ν+ 3)u00 = 0 (4.2)
Integrating Eq.(4.2) once with respect toξ and setting the integration constant as zero yields
u000+ 3(u0)2+ (4ν+ 3)u0= 0 (4.3)
Settingu0=v, Eq.(4.3) becomes
V00+ 3V2+ (4ν+ 3)V = 0 (4.4)
With balancing according procedure that be described, we get n = 2, therefore the solution of Eq.(4.1) can be expressed as follows:
v(ξ) = 2
X
i=0
Ai(G(ξ))i (4.5)
whereG(ξ) satisfies the Riccati equation andA0, A1, A2 are parameters to be determined. solutions of Riccati equation are given in (2.4), (2.5). With substituting (4.5) into (4.4) and use of (2.3) and then equating all coefficients of the functionsGi to zero, we obtain A0,A1 and A2:
Case 1 : A0 =− c2
3, A1=−2cd, A2 =−2d
2; acd6= 0 (4.6)
Case 2 : A0 = 0, A1=−2cd, A2 =−2d2; acd6= 0 (4.7)
Recall u0(x, y, z, t) = v(x, y, z, t), therefore when d < 0 and c > 0 the solutions of Eq.(4.1) with using Case 1 (4.6) is given by
u1,2(x, y, z, t) =−
c2(x+y+z−ν1,2t)
3 −
2c
1−dexp[c(x+y+z−ν1,2t+x1)] ,
whereν1,2 = 8(−3± 16c + 9) andx1 is a constant of integration. also solutions of Eq.(4.1) with using Case 2 (4.7) is given by
u3,4(x, y, z, t) =−
2c
1−dexp[c(x+y+z−ν3,4t+x2)]
(4.9)
whereν3,4 = 18(−3± √
9−16c2) andx
2 is a constant of integration. And when d >0 andc <0 the solutions of Eq.(4.1) with using Case 1 (4.6) is given by
u5,6(x, y, z, t) =−
c2(x+y+z−ν1,2t)
3 −
2c
1 +dexp[c(x+y+z−ν1,2t+x3)] ,
(4.10) wherex3 is a constant of integration.
also solutions of Eq.(4.1) with using Case 2 (4.7) is given by
u7,8(x, y, z, t) =−
2c
1 +dexp[c(x+y+z−ν3,4t+x4)]
(4.11)
wherex4 is a constant of integration.
5. Concluding remarks
In this paper, the simplest equation method has been successfully used to obtain exact solutions of the Jimbo-Miwa equation and Potential YTSF equation. As the simplest equations, we have used the equation of Riccati . For this simplest equation, we have obtained a balance equation. By means of balance equations, we obtained exact solutions of the studied class nonlinear PDEs. We have also verified that these solutions we have found are indeed solutions to the original equations.
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